Problem: All of the 3rd grade teachers and students from Covington went on a field trip to an archaeology museum. Tickets were $$5.00$ each for teachers and $$3.50$ each for students, and the group paid $$48.00$ in total. The next month, the same group visited a science museum where the tickets cost $$15.00$ each for teachers and $$9.50$ each for students, and the group paid $$136.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+3.5y = 48}$ ${15x+9.5y = 136}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-15x-10.5y = -144}$ ${15x+9.5y = 136}$ Add the top and bottom equations together. $ -y = -8 $ $ y = \dfrac{-8}{-1}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {5x+3.5y = 48}$ to find $x$ ${5x + 3.5}{(8)}{= 48}$ $5x+28 = 48$ $5x = 20$ $x = \dfrac{20}{5}$ ${x = 4}$ You can also plug ${y = 8}$ into $ {15x+9.5y = 136}$ and get the same answer for $x$ ${15x + 9.5}{(8)}{= 136}$ ${x = 4}$ There were $4$ teachers and $8$ students on the field trips.